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]

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

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {36, 31} \[ \frac {\log (b+x)}{a-b}-\frac {\log (a+x)}{a-b} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + x)*(b + x)),x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rubi steps

\begin {align*} \int \frac {1}{(a+x) (b+x)} \, dx &=\frac {\int \frac {1}{a+x} \, dx}{-a+b}-\frac {\int \frac {1}{b+x} \, dx}{-a+b}\\ &=-\frac {\log (a+x)}{a-b}+\frac {\log (b+x)}{a-b}\\ \end {align*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.40, size = 20, normalized size = 0.77 \[ -\frac {\log \left (a + x\right ) - \log \left (b + x\right )}{a - b} \]

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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giac [A]  time = 0.91, size = 28, normalized size = 1.08 \[ -\frac {\log \left ({\left | a + x \right |}\right )}{a - b} + \frac {\log \left ({\left | b + x \right |}\right )}{a - b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+x)/(b+x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 27, normalized size = 1.04 \[ -\frac {\ln \left (a +x \right )}{a -b}+\frac {\ln \left (b +x \right )}{a -b} \]

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 = 0.59, size = 26, normalized size = 1.00 \[ -\frac {\log \left (a + x\right )}{a - b} + \frac {\log \left (b + x\right )}{a - b} \]

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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mupad [B]  time = 0.22, size = 18, normalized size = 0.69 \[ \frac {\ln \left (\frac {b+x}{a+x}\right )}{a-b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + x)*(b + x)),x)

[Out]

log((b + x)/(a + x))/(a - b)

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sympy [B]  time = 0.23, 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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