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

Optimal. Leaf size=65 \[ \frac{1}{2} \sqrt{x^2+1} x+\sqrt{x^2+1}+\frac{\sqrt{x^2+1}}{x}-\log \left (\sqrt{x^2+1}+1\right )-x-\frac{1}{x}-\frac{1}{2} \sinh ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.255223, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{1}{2} \sqrt{x^2+1} x+\sqrt{x^2+1}+\frac{\sqrt{x^2+1}}{x}-\log \left (\sqrt{x^2+1}+1\right )-x-\frac{1}{x}-\frac{1}{2} \sinh ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0436108, size = 49, normalized size = 0.75 \[ \sqrt{x^2+1} \left (\frac{x}{2}+\frac{1}{x}+1\right )-\log \left (\sqrt{x^2+1}+1\right )-x-\frac{1}{x}-\frac{1}{2} \sinh ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 56, normalized size = 0.9 \[ -x-{x}^{-1}-{\frac{x}{2}\sqrt{{x}^{2}+1}}-{\frac{{\it Arcsinh} \left ( x \right ) }{2}}+\sqrt{{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\sqrt{{x}^{2}+1}}} \right ) -\ln \left ( x \right ) +{\frac{1}{x} \left ({x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ 2 \, x - 5 \, \arctan \left (\frac{1}{2} \, x\right ) + \int \frac{x^{6} + x^{5} - x^{4}}{3 \, x^{4} + 16 \, x^{2} +{\left (x^{4} + 8 \, x^{2} + 16\right )} \sqrt{x^{2} + 1} + 16}\,{d x} + \log \left (x^{2} + 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x - 5*arctan(1/2*x) + integrate((x^6 + x^5 - x^4)/(3*x^4 + 16*x^2 + (x^4 + 8*x
^2 + 16)*sqrt(x^2 + 1) + 16), x) + log(x^2 + 4)

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Fricas [A]  time = 0.281266, size = 332, normalized size = 5.11 \[ -\frac{4 \, x^{6} + 16 \, x^{5} + 5 \, x^{4} + 24 \, x^{3} + 5 \, x^{2} + 2 \,{\left (4 \, x^{4} + 3 \, x^{2}\right )} \log \left (x\right ) + 2 \,{\left (4 \, x^{4} + 3 \, x^{2} -{\left (4 \, x^{3} + x\right )} \sqrt{x^{2} + 1}\right )} \log \left (-x + \sqrt{x^{2} + 1} + 1\right ) -{\left (4 \, x^{4} + 3 \, x^{2} -{\left (4 \, x^{3} + x\right )} \sqrt{x^{2} + 1}\right )} \log \left (-x + \sqrt{x^{2} + 1}\right ) - 2 \,{\left (4 \, x^{4} + 3 \, x^{2} -{\left (4 \, x^{3} + x\right )} \sqrt{x^{2} + 1}\right )} \log \left (-x + \sqrt{x^{2} + 1} - 1\right ) -{\left (4 \, x^{5} + 16 \, x^{4} + 3 \, x^{3} + 16 \, x^{2} + 2 \,{\left (4 \, x^{3} + x\right )} \log \left (x\right ) + 4 \, x + 2\right )} \sqrt{x^{2} + 1} + 8 \, x + 2}{2 \,{\left (4 \, x^{4} + 3 \, x^{2} -{\left (4 \, x^{3} + x\right )} \sqrt{x^{2} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(4*x^6 + 16*x^5 + 5*x^4 + 24*x^3 + 5*x^2 + 2*(4*x^4 + 3*x^2)*log(x) + 2*(4*
x^4 + 3*x^2 - (4*x^3 + x)*sqrt(x^2 + 1))*log(-x + sqrt(x^2 + 1) + 1) - (4*x^4 +
3*x^2 - (4*x^3 + x)*sqrt(x^2 + 1))*log(-x + sqrt(x^2 + 1)) - 2*(4*x^4 + 3*x^2 -
(4*x^3 + x)*sqrt(x^2 + 1))*log(-x + sqrt(x^2 + 1) - 1) - (4*x^5 + 16*x^4 + 3*x^3
+ 16*x^2 + 2*(4*x^3 + x)*log(x) + 4*x + 2)*sqrt(x^2 + 1) + 8*x + 2)/(4*x^4 + 3*
x^2 - (4*x^3 + x)*sqrt(x^2 + 1))

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Sympy [A]  time = 11.8625, size = 76, normalized size = 1.17 \[ \frac{x \sqrt{x^{2} + 1}}{2} - x + \frac{x}{\sqrt{x^{2} + 1}} + \sqrt{x^{2} + 1} - \log{\left (1 + \frac{1}{\sqrt{x^{2} + 1}} \right )} + \log{\left (\frac{1}{\sqrt{x^{2} + 1}} \right )} - \frac{\operatorname{asinh}{\left (x \right )}}{2} - \frac{1}{x} + \frac{1}{x \sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.273024, size = 120, normalized size = 1.85 \[ \frac{1}{2} \, \sqrt{x^{2} + 1}{\left (x + 2\right )} - x - \frac{2}{{\left (x - \sqrt{x^{2} + 1}\right )}^{2} - 1} - \frac{1}{x} + \frac{1}{2} \,{\rm ln}\left (-x + \sqrt{x^{2} + 1}\right ) -{\rm ln}\left ({\left | x \right |}\right ) -{\rm ln}\left ({\left | -x + \sqrt{x^{2} + 1} + 1 \right |}\right ) +{\rm ln}\left ({\left | -x + \sqrt{x^{2} + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(x^2 + 1)*(x + 2) - x - 2/((x - sqrt(x^2 + 1))^2 - 1) - 1/x + 1/2*ln(-x
+ sqrt(x^2 + 1)) - ln(abs(x)) - ln(abs(-x + sqrt(x^2 + 1) + 1)) + ln(abs(-x + sq
rt(x^2 + 1) - 1))