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

Optimal. Leaf size=53 \[ \frac{1}{12} \left (2 x^3+6 x^2+\left (-2 x^2-3 x+4\right ) \sqrt{x^2+1}-6 \log \left (\sqrt{x^2+1}+1\right )-3 \sinh ^{-1}(x)\right ) \]

[Out]

(6*x^2 + 2*x^3 + (4 - 3*x - 2*x^2)*Sqrt[1 + x^2] - 3*ArcSinh[x] - 6*Log[1 + Sqrt
[1 + x^2]])/12

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Rubi [A]  time = 0.365495, antiderivative size = 101, normalized size of antiderivative = 1.91, number of steps used = 12, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{x^3}{6}+\frac{x^2}{2}-\frac{1}{4} \sqrt{x^2+1} x-\frac{1}{6} \left (x^2+1\right )^{3/2}+\frac{1}{2 \left (\sqrt{x^2+1}+x\right )}+\frac{1}{2} \log \left (\sqrt{x^2+1}+x\right )-\log \left (\sqrt{x^2+1}+x+1\right )+\frac{x}{2}-\frac{1}{4} \sinh ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

x/2 + x^2/2 + x^3/6 - (x*Sqrt[1 + x^2])/4 - (1 + x^2)^(3/2)/6 + 1/(2*(x + Sqrt[1
+ x^2])) - ArcSinh[x]/4 + Log[x + Sqrt[1 + x^2]]/2 - Log[1 + x + Sqrt[1 + x^2]]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0489116, size = 53, normalized size = 1. \[ \frac{1}{12} \left (2 x^3+6 x^2+\left (-2 x^2-3 x+4\right ) \sqrt{x^2+1}-6 \log \left (\sqrt{x^2+1}+1\right )-3 \sinh ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(6*x^2 + 2*x^3 + (4 - 3*x - 2*x^2)*Sqrt[1 + x^2] - 3*ArcSinh[x] - 6*Log[1 + Sqrt
[1 + x^2]])/12

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Maple [A]  time = 0.007, size = 58, normalized size = 1.1 \[{\frac{{x}^{2}}{2}}-{\frac{\ln \left ( x \right ) }{2}}+{\frac{{x}^{3}}{6}}-{\frac{x}{4}\sqrt{{x}^{2}+1}}-{\frac{{\it Arcsinh} \left ( x \right ) }{4}}+{\frac{1}{2}\sqrt{{x}^{2}+1}}-{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{{x}^{2}+1}}} \right ) }-{\frac{1}{6} \left ({x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2-1/2*ln(x)+1/6*x^3-1/4*x*(x^2+1)^(1/2)-1/4*arcsinh(x)+1/2*(x^2+1)^(1/2)-1
/2*arctanh(1/(x^2+1)^(1/2))-1/6*(x^2+1)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{4} \, x^{2} - \frac{3}{56} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (4 \, x + 3\right )}\right ) + \frac{1}{4} \, x + \int \frac{x^{4} + x^{3} - x^{2}}{4 \, x^{5} + 12 \, x^{4} + 19 \, x^{3} + 19 \, x^{2} +{\left (4 \, x^{4} + 12 \, x^{3} + 17 \, x^{2} + 12 \, x + 4\right )} \sqrt{x^{2} + 1} + 12 \, x + 4}\,{d x} - \frac{7}{16} \, \log \left (2 \, x^{2} + 3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x^2 - 3/56*sqrt(7)*arctan(1/7*sqrt(7)*(4*x + 3)) + 1/4*x + integrate((x^4 +
x^3 - x^2)/(4*x^5 + 12*x^4 + 19*x^3 + 19*x^2 + (4*x^4 + 12*x^3 + 17*x^2 + 12*x +
4)*sqrt(x^2 + 1) + 12*x + 4), x) - 7/16*log(2*x^2 + 3*x + 2)

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Fricas [A]  time = 0.280396, size = 311, normalized size = 5.87 \[ \frac{16 \, x^{6} + 36 \, x^{5} + 33 \, x^{3} - 18 \, x^{2} - 6 \,{\left (4 \, x^{3} + 3 \, x\right )} \log \left (x\right ) - 6 \,{\left (4 \, x^{3} -{\left (4 \, x^{2} + 1\right )} \sqrt{x^{2} + 1} + 3 \, x\right )} \log \left (-x + \sqrt{x^{2} + 1} + 1\right ) + 3 \,{\left (4 \, x^{3} -{\left (4 \, x^{2} + 1\right )} \sqrt{x^{2} + 1} + 3 \, x\right )} \log \left (-x + \sqrt{x^{2} + 1}\right ) + 6 \,{\left (4 \, x^{3} -{\left (4 \, x^{2} + 1\right )} \sqrt{x^{2} + 1} + 3 \, x\right )} \log \left (-x + \sqrt{x^{2} + 1} - 1\right ) -{\left (16 \, x^{5} + 36 \, x^{4} - 8 \, x^{3} + 15 \, x^{2} - 6 \,{\left (4 \, x^{2} + 1\right )} \log \left (x\right ) - 12 \, x\right )} \sqrt{x^{2} + 1} + 3 \, x - 4}{12 \,{\left (4 \, x^{3} -{\left (4 \, x^{2} + 1\right )} \sqrt{x^{2} + 1} + 3 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(16*x^6 + 36*x^5 + 33*x^3 - 18*x^2 - 6*(4*x^3 + 3*x)*log(x) - 6*(4*x^3 - (4
*x^2 + 1)*sqrt(x^2 + 1) + 3*x)*log(-x + sqrt(x^2 + 1) + 1) + 3*(4*x^3 - (4*x^2 +
1)*sqrt(x^2 + 1) + 3*x)*log(-x + sqrt(x^2 + 1)) + 6*(4*x^3 - (4*x^2 + 1)*sqrt(x
^2 + 1) + 3*x)*log(-x + sqrt(x^2 + 1) - 1) - (16*x^5 + 36*x^4 - 8*x^3 + 15*x^2 -
6*(4*x^2 + 1)*log(x) - 12*x)*sqrt(x^2 + 1) + 3*x - 4)/(4*x^3 - (4*x^2 + 1)*sqrt
(x^2 + 1) + 3*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.270023, size = 108, normalized size = 2.04 \[ \frac{1}{6} \, x^{3} + \frac{1}{2} \, x^{2} - \frac{1}{12} \,{\left ({\left (2 \, x + 3\right )} x - 4\right )} \sqrt{x^{2} + 1} + \frac{1}{4} \,{\rm ln}\left (-x + \sqrt{x^{2} + 1}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | x \right |}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} + 1} + 1 \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x^3 + 1/2*x^2 - 1/12*((2*x + 3)*x - 4)*sqrt(x^2 + 1) + 1/4*ln(-x + sqrt(x^2
+ 1)) - 1/2*ln(abs(x)) - 1/2*ln(abs(-x + sqrt(x^2 + 1) + 1)) + 1/2*ln(abs(-x + s
qrt(x^2 + 1) - 1))