3.2428 \(\int \left (\frac{1}{x}-\frac{1}{x \sqrt{1+b x+c x^2}}\right ) \, dx\)

Optimal. Leaf size=23 \[ \log \left (-2 \sqrt{b x+c x^2+1}-b x-2\right ) \]

[Out]

Log[-2 - b*x - 2*Sqrt[1 + b*x + c*x^2]]

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Rubi [A]  time = 0.0483027, antiderivative size = 27, normalized size of antiderivative = 1.17, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \tanh ^{-1}\left (\frac{b x+2}{2 \sqrt{b x+c x^2+1}}\right )+\log (x) \]

Antiderivative was successfully verified.

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

[Out]

ArcTanh[(2 + b*x)/(2*Sqrt[1 + b*x + c*x^2])] + Log[x]

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Rubi in Sympy [A]  time = 6.23263, size = 24, normalized size = 1.04 \[ \log{\left (x \right )} + \operatorname{atanh}{\left (\frac{b x + 2}{2 \sqrt{b x + c x^{2} + 1}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x) + atanh((b*x + 2)/(2*sqrt(b*x + c*x**2 + 1)))

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Mathematica [A]  time = 0.0523502, size = 22, normalized size = 0.96 \[ \log \left (2 \sqrt{b x+c x^2+1}+b x+2\right ) \]

Antiderivative was successfully verified.

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

[Out]

Log[2 + b*x + 2*Sqrt[1 + b*x + c*x^2]]

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Maple [A]  time = 0.006, size = 24, normalized size = 1. \[{\it Artanh} \left ({\frac{bx+2}{2}{\frac{1}{\sqrt{c{x}^{2}+bx+1}}}} \right ) +\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctanh(1/2*(b*x+2)/(c*x^2+b*x+1)^(1/2))+ln(x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224961, size = 41, normalized size = 1.78 \[ \log \left (x\right ) - \log \left (-\frac{b x - 2 \, \sqrt{c x^{2} + b x + 1} + 2}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x) - log(-(b*x - 2*sqrt(c*x^2 + b*x + 1) + 2)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x + c x^{2} + 1} - 1}{x \sqrt{b x + c x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.237438, size = 68, normalized size = 2.96 \[{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + b x + 1} + 1 \right |}\right ) -{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + b x + 1} - 1 \right |}\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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