Optimal. Leaf size=18 \[ \frac {\log (x)}{a}-\frac {\log (a+b x)}{a} \]
________________________________________________________________________________________
Rubi [A] time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {36, 29, 31} \begin {gather*} \frac {\log (x)}{a}-\frac {\log (a+b x)}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rubi steps
\begin {align*} \int \frac {1}{x (a+b x)} \, dx &=\frac {\int \frac {1}{x} \, dx}{a}-\frac {b \int \frac {1}{a+b x} \, dx}{a}\\ &=\frac {\log (x)}{a}-\frac {\log (a+b x)}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {\log (x)}{a}-\frac {\log (a+b x)}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x (a+b x)} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.25, size = 16, normalized size = 0.89 \begin {gather*} -\frac {\log \left (b x + a\right ) - \log \relax (x)}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.10, size = 20, normalized size = 1.11 \begin {gather*} -\frac {\log \left ({\left | b x + a \right |}\right )}{a} + \frac {\log \left ({\left | x \right |}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 19, normalized size = 1.06 \begin {gather*} \frac {\ln \relax (x )}{a}-\frac {\ln \left (b x +a \right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.39, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\log \left (b x + a\right )}{a} + \frac {\log \relax (x)}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.09, size = 15, normalized size = 0.83 \begin {gather*} -\frac {2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.15, size = 10, normalized size = 0.56 \begin {gather*} \frac {\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________