3.6 \(\int \frac{x}{(1+x) (2+x) (3+x)} \, dx\)

Optimal. Leaf size=23 \[ -\frac{1}{2} \log (x+1)+2 \log (x+2)-\frac{3}{2} \log (x+3) \]

[Out]

-Log[1 + x]/2 + 2*Log[2 + x] - (3*Log[3 + x])/2

_______________________________________________________________________________________

Rubi [A]  time = 0.0438025, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{1}{2} \log (x+1)+2 \log (x+2)-\frac{3}{2} \log (x+3) \]

Antiderivative was successfully verified.

[In]  Int[x/((1 + x)*(2 + x)*(3 + x)),x]

[Out]

-Log[1 + x]/2 + 2*Log[2 + x] - (3*Log[3 + x])/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.41681, size = 20, normalized size = 0.87 \[ - \frac{\log{\left (x + 1 \right )}}{2} + 2 \log{\left (x + 2 \right )} - \frac{3 \log{\left (x + 3 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(1+x)/(2+x)/(3+x),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0114391, size = 23, normalized size = 1. \[ -\frac{1}{2} \log (x+1)+2 \log (x+2)-\frac{3}{2} \log (x+3) \]

Antiderivative was successfully verified.

[In]  Integrate[x/((1 + x)*(2 + x)*(3 + x)),x]

[Out]

-Log[1 + x]/2 + 2*Log[2 + x] - (3*Log[3 + x])/2

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 20, normalized size = 0.9 \[ -{\frac{\ln \left ( 1+x \right ) }{2}}+2\,\ln \left ( 2+x \right ) -{\frac{3\,\ln \left ( 3+x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(1+x)/(2+x)/(3+x),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.35939, size = 26, normalized size = 1.13 \[ -\frac{3}{2} \, \log \left (x + 3\right ) + 2 \, \log \left (x + 2\right ) - \frac{1}{2} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x + 3)*(x + 2)*(x + 1)),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.259598, size = 26, normalized size = 1.13 \[ -\frac{3}{2} \, \log \left (x + 3\right ) + 2 \, \log \left (x + 2\right ) - \frac{1}{2} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x + 3)*(x + 2)*(x + 1)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.285438, size = 20, normalized size = 0.87 \[ - \frac{\log{\left (x + 1 \right )}}{2} + 2 \log{\left (x + 2 \right )} - \frac{3 \log{\left (x + 3 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(1+x)/(2+x)/(3+x),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214715, size = 30, normalized size = 1.3 \[ -\frac{3}{2} \,{\rm ln}\left ({\left | x + 3 \right |}\right ) + 2 \,{\rm ln}\left ({\left | x + 2 \right |}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x + 3)*(x + 2)*(x + 1)),x, algorithm="giac")

[Out]

-3/2*ln(abs(x + 3)) + 2*ln(abs(x + 2)) - 1/2*ln(abs(x + 1))