3.179 \(\int \frac{1}{(1+x) (2+x)} \, dx\)

Optimal. Leaf size=11 \[ \log (x+1)-\log (x+2) \]

[Out]

Log[1 + x] - Log[2 + x]

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Rubi [A]  time = 0.00801813, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \log (x+1)-\log (x+2) \]

Antiderivative was successfully verified.

[In]  Int[1/((1 + x)*(2 + x)),x]

[Out]

Log[1 + x] - Log[2 + x]

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Rubi in Sympy [A]  time = 1.05143, size = 8, normalized size = 0.73 \[ \log{\left (x + 1 \right )} - \log{\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1+x)/(2+x),x)

[Out]

log(x + 1) - log(x + 2)

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Mathematica [A]  time = 0.00433449, size = 11, normalized size = 1. \[ \log (x+1)-\log (x+2) \]

Antiderivative was successfully verified.

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

[Out]

Log[1 + x] - Log[2 + x]

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Maple [A]  time = 0.007, size = 12, normalized size = 1.1 \[ \ln \left ( 1+x \right ) -\ln \left ( 2+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x)/(2+x),x)

[Out]

ln(1+x)-ln(2+x)

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Maxima [A]  time = 1.39602, size = 15, normalized size = 1.36 \[ -\log \left (x + 2\right ) + \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x + 2) + log(x + 1)

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Fricas [A]  time = 0.195656, size = 15, normalized size = 1.36 \[ -\log \left (x + 2\right ) + \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x + 2) + log(x + 1)

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Sympy [A]  time = 0.089869, size = 8, normalized size = 0.73 \[ \log{\left (x + 1 \right )} - \log{\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+x)/(2+x),x)

[Out]

log(x + 1) - log(x + 2)

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GIAC/XCAS [A]  time = 0.206493, size = 18, normalized size = 1.64 \[ -{\rm ln}\left ({\left | x + 2 \right |}\right ) +{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(abs(x + 2)) + ln(abs(x + 1))