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

Optimal. Leaf size=46 \[ \frac{\sqrt{x^2+1}}{x}+\sqrt{x^2+1}-\log \left (\sqrt{x^2+1}+1\right )-\frac{1}{x}-\sinh ^{-1}(x) \]

[Out]

-x^(-1) + Sqrt[1 + x^2] + Sqrt[1 + x^2]/x - ArcSinh[x] - Log[1 + Sqrt[1 + x^2]]

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Rubi [A]  time = 0.145907, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ \frac{\sqrt{x^2+1}}{x}+\sqrt{x^2+1}-\log \left (\sqrt{x^2+1}+1\right )-\frac{1}{x}-\sinh ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

-x^(-1) + Sqrt[1 + x^2] + Sqrt[1 + x^2]/x - ArcSinh[x] - Log[1 + Sqrt[1 + x^2]]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0405428, size = 39, normalized size = 0.85 \[ \sqrt{x^2+1} \left (\frac{1}{x}+1\right )-\log \left (\sqrt{x^2+1}+1\right )-\frac{1}{x}-\sinh ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

-x^(-1) + (1 + x^(-1))*Sqrt[1 + x^2] - ArcSinh[x] - Log[1 + Sqrt[1 + x^2]]

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Maple [A]  time = 0.005, size = 53, normalized size = 1.2 \[ -{x}^{-1}+\sqrt{{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\sqrt{{x}^{2}+1}}} \right ) -\ln \left ( x \right ) +{\frac{1}{x} \left ({x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-x\sqrt{{x}^{2}+1}-{\it Arcsinh} \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-1+x)/(1+(x^2+1)^(1/2)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{4} \, x^{2} - \frac{1}{2} \, x - \int \frac{x^{3} - x^{2}}{2 \,{\left (x^{2} + 2 \, \sqrt{x^{2} + 1} + 2\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.27678, size = 201, normalized size = 4.37 \[ -\frac{2 \, x^{4} + 4 \, x^{2} -{\left (2 \, x^{3} - 2 \, \sqrt{x^{2} + 1} x^{2} + x\right )} \log \left (2 \, x^{2} - \sqrt{x^{2} + 1}{\left (2 \, x + 1\right )} + x + 1\right ) +{\left (2 \, x^{3} + x\right )} \log \left (x\right ) +{\left (2 \, x^{3} - 2 \, \sqrt{x^{2} + 1} x^{2} + x\right )} \log \left (-x + \sqrt{x^{2} + 1} + 1\right ) -{\left (2 \, x^{3} + 2 \, x^{2} \log \left (x\right ) + 3 \, x + 1\right )} \sqrt{x^{2} + 1} + x + 1}{2 \, x^{3} - 2 \, \sqrt{x^{2} + 1} x^{2} + x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.15092, size = 61, normalized size = 1.33 \[ \frac{x}{\sqrt{x^{2} + 1}} + \sqrt{x^{2} + 1} - \log{\left (1 + \frac{1}{\sqrt{x^{2} + 1}} \right )} + \log{\left (\frac{1}{\sqrt{x^{2} + 1}} \right )} - \operatorname{asinh}{\left (x \right )} - \frac{1}{x} + \frac{1}{x \sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/sqrt(x**2 + 1) + sqrt(x**2 + 1) - log(1 + 1/sqrt(x**2 + 1)) + log(1/sqrt(x**2
+ 1)) - asinh(x) - 1/x + 1/(x*sqrt(x**2 + 1))

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GIAC/XCAS [A]  time = 0.266822, size = 107, normalized size = 2.33 \[ \sqrt{x^{2} + 1} - \frac{2}{{\left (x - \sqrt{x^{2} + 1}\right )}^{2} - 1} - \frac{1}{x} +{\rm ln}\left (-x + \sqrt{x^{2} + 1}\right ) -{\rm ln}\left ({\left | x \right |}\right ) -{\rm ln}\left ({\left | -x + \sqrt{x^{2} + 1} + 1 \right |}\right ) +{\rm ln}\left ({\left | -x + \sqrt{x^{2} + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x^2 + 1) - 2/((x - sqrt(x^2 + 1))^2 - 1) - 1/x + ln(-x + sqrt(x^2 + 1)) - l
n(abs(x)) - ln(abs(-x + sqrt(x^2 + 1) + 1)) + ln(abs(-x + sqrt(x^2 + 1) - 1))