3.181 \(\int \frac{1}{(a+x) (b+x)} \, dx\)

Optimal. Leaf size=26 \[ \frac{\log (b+x)}{a-b}-\frac{\log (a+x)}{a-b} \]

[Out]

-(Log[a + x]/(a - b)) + Log[b + x]/(a - b)

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Rubi [A]  time = 0.0205634, antiderivative size = 26, 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 \[ \frac{\log (b+x)}{a-b}-\frac{\log (a+x)}{a-b} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + x)*(b + x)),x]

[Out]

-(Log[a + x]/(a - b)) + Log[b + x]/(a - b)

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Rubi in Sympy [A]  time = 1.79153, size = 15, normalized size = 0.58 \[ - \frac{\log{\left (a + x \right )}}{a - b} + \frac{\log{\left (b + x \right )}}{a - b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+x)/(b+x),x)

[Out]

-log(a + x)/(a - b) + log(b + x)/(a - b)

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Mathematica [A]  time = 0.00975916, size = 19, normalized size = 0.73 \[ \frac{\log (b+x)-\log (a+x)}{a-b} \]

Antiderivative was successfully verified.

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

[Out]

(-Log[a + x] + Log[b + x])/(a - b)

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Maple [A]  time = 0.009, size = 27, normalized size = 1. \[ -{\frac{\ln \left ( a+x \right ) }{a-b}}+{\frac{\ln \left ( b+x \right ) }{a-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+x)/(b+x),x)

[Out]

-ln(a+x)/(a-b)+ln(b+x)/(a-b)

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Maxima [A]  time = 1.36114, size = 35, normalized size = 1.35 \[ -\frac{\log \left (a + x\right )}{a - b} + \frac{\log \left (b + x\right )}{a - b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(a + x)/(a - b) + log(b + x)/(a - b)

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Fricas [A]  time = 0.221043, size = 27, normalized size = 1.04 \[ -\frac{\log \left (a + x\right ) - \log \left (b + x\right )}{a - b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(log(a + x) - log(b + x))/(a - b)

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Sympy [A]  time = 0.28664, size = 80, normalized size = 3.08 \[ \frac{\log{\left (- \frac{a^{2}}{2 \left (a - b\right )} + \frac{a b}{a - b} + \frac{a}{2} - \frac{b^{2}}{2 \left (a - b\right )} + \frac{b}{2} + x \right )}}{a - b} - \frac{\log{\left (\frac{a^{2}}{2 \left (a - b\right )} - \frac{a b}{a - b} + \frac{a}{2} + \frac{b^{2}}{2 \left (a - b\right )} + \frac{b}{2} + x \right )}}{a - b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+x)/(b+x),x)

[Out]

log(-a**2/(2*(a - b)) + a*b/(a - b) + a/2 - b**2/(2*(a - b)) + b/2 + x)/(a - b)
- log(a**2/(2*(a - b)) - a*b/(a - b) + a/2 + b**2/(2*(a - b)) + b/2 + x)/(a - b)

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GIAC/XCAS [A]  time = 0.21192, size = 38, normalized size = 1.46 \[ -\frac{{\rm ln}\left ({\left | a + x \right |}\right )}{a - b} + \frac{{\rm ln}\left ({\left | b + x \right |}\right )}{a - b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(abs(a + x))/(a - b) + ln(abs(b + x))/(a - b)