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

Optimal. Leaf size=21 \[ -\frac{x^3}{3}-\frac{1}{3} \left (x^2+1\right )^{3/2} \]

[Out]

-x^3/3 - (1 + x^2)^(3/2)/3

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Rubi [A]  time = 0.0403636, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{x^3}{3}-\frac{1}{3} \left (x^2+1\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x/(x - Sqrt[1 + x^2]),x]

[Out]

-x^3/3 - (1 + x^2)^(3/2)/3

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Rubi in Sympy [A]  time = 2.57951, size = 15, normalized size = 0.71 \[ - \frac{x^{3}}{3} - \frac{\left (x^{2} + 1\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**3/3 - (x**2 + 1)**(3/2)/3

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Mathematica [A]  time = 0.0216363, size = 21, normalized size = 1. \[ \frac{1}{3} \left (-x^3-\left (x^2+1\right )^{3/2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-x^3 - (1 + x^2)^(3/2))/3

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Maple [A]  time = 0.004, size = 16, normalized size = 0.8 \[ -{\frac{{x}^{3}}{3}}-{\frac{1}{3} \left ({x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x-(x^2+1)^(1/2)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(x - sqrt(x^2 + 1)), x)

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Fricas [A]  time = 0.26072, size = 77, normalized size = 3.67 \[ \frac{6 \, x^{4} + 6 \, x^{2} - 3 \,{\left (2 \, x^{3} + x\right )} \sqrt{x^{2} + 1} + 1}{3 \,{\left (4 \, x^{3} -{\left (4 \, x^{2} + 1\right )} \sqrt{x^{2} + 1} + 3 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(6*x^4 + 6*x^2 - 3*(2*x^3 + x)*sqrt(x^2 + 1) + 1)/(4*x^3 - (4*x^2 + 1)*sqrt(
x^2 + 1) + 3*x)

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Sympy [A]  time = 1.49251, size = 56, normalized size = 2.67 \[ \frac{2 x^{2}}{3 x - 3 \sqrt{x^{2} + 1}} - \frac{x \sqrt{x^{2} + 1}}{3 x - 3 \sqrt{x^{2} + 1}} + \frac{1}{3 x - 3 \sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.261396, size = 20, normalized size = 0.95 \[ -\frac{1}{3} \, x^{3} - \frac{1}{3} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*x^3 - 1/3*(x^2 + 1)^(3/2)