3.31 \(\int \frac{1+x}{\left (4+2 x+x^2\right ) \sqrt{5+2 x+x^2}} \, dx\)

Optimal. Leaf size=15 \[ -\tanh ^{-1}\left (\sqrt{x^2+2 x+5}\right ) \]

[Out]

-ArcTanh[Sqrt[5 + 2*x + x^2]]

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Rubi [A]  time = 0.0524999, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\tanh ^{-1}\left (\sqrt{x^2+2 x+5}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)/((4 + 2*x + x^2)*Sqrt[5 + 2*x + x^2]),x]

[Out]

-ArcTanh[Sqrt[5 + 2*x + x^2]]

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Rubi in Sympy [A]  time = 13.311, size = 14, normalized size = 0.93 \[ - \operatorname{atanh}{\left (\sqrt{x^{2} + 2 x + 5} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)/(x**2+2*x+4)/(x**2+2*x+5)**(1/2),x)

[Out]

-atanh(sqrt(x**2 + 2*x + 5))

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Mathematica [B]  time = 0.0178435, size = 41, normalized size = 2.73 \[ \frac{1}{2} \log \left (1-\sqrt{x^2+2 x+5}\right )-\frac{1}{2} \log \left (\sqrt{x^2+2 x+5}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x)/((4 + 2*x + x^2)*Sqrt[5 + 2*x + x^2]),x]

[Out]

Log[1 - Sqrt[5 + 2*x + x^2]]/2 - Log[1 + Sqrt[5 + 2*x + x^2]]/2

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Maple [A]  time = 0.011, size = 14, normalized size = 0.9 \[ -{\it Artanh} \left ( \sqrt{{x}^{2}+2\,x+5} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)/(x^2+2*x+4)/(x^2+2*x+5)^(1/2),x)

[Out]

-arctanh((x^2+2*x+5)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(sqrt(x^2 + 2*x + 5)*(x^2 + 2*x + 4)),x, algorithm="maxima")

[Out]

integrate((x + 1)/(sqrt(x^2 + 2*x + 5)*(x^2 + 2*x + 4)), x)

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Fricas [A]  time = 0.272681, size = 66, normalized size = 4.4 \[ \frac{1}{2} \, \log \left (x^{2} - \sqrt{x^{2} + 2 \, x + 5}{\left (x + 2\right )} + 3 \, x + 6\right ) - \frac{1}{2} \, \log \left (x^{2} - \sqrt{x^{2} + 2 \, x + 5} x + x + 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(sqrt(x^2 + 2*x + 5)*(x^2 + 2*x + 4)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.39663, size = 36, normalized size = 2.4 \[ \frac{\log{\left (-1 + \frac{1}{\sqrt{x^{2} + 2 x + 5}} \right )}}{2} - \frac{\log{\left (1 + \frac{1}{\sqrt{x^{2} + 2 x + 5}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)/(x**2+2*x+4)/(x**2+2*x+5)**(1/2),x)

[Out]

log(-1 + 1/sqrt(x**2 + 2*x + 5))/2 - log(1 + 1/sqrt(x**2 + 2*x + 5))/2

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GIAC/XCAS [A]  time = 0.276584, size = 78, normalized size = 5.2 \[ \frac{1}{2} \,{\rm ln}\left ({\left (x - \sqrt{x^{2} + 2 \, x + 5}\right )}^{2} + 4 \, x - 4 \, \sqrt{x^{2} + 2 \, x + 5} + 7\right ) - \frac{1}{2} \,{\rm ln}\left ({\left (x - \sqrt{x^{2} + 2 \, x + 5}\right )}^{2} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(sqrt(x^2 + 2*x + 5)*(x^2 + 2*x + 4)),x, algorithm="giac")

[Out]

1/2*ln((x - sqrt(x^2 + 2*x + 5))^2 + 4*x - 4*sqrt(x^2 + 2*x + 5) + 7) - 1/2*ln((
x - sqrt(x^2 + 2*x + 5))^2 + 3)