Optimal. Leaf size=36 \[ \frac {\log (a+b x)}{b c-a d}-\frac {\log (c+d x)}{b c-a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {36, 31} \[ \frac {\log (a+b x)}{b c-a d}-\frac {\log (c+d x)}{b c-a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 36
Rubi steps
\begin {align*} \int \frac {1}{(a+b x) (c+d x)} \, dx &=\frac {b \int \frac {1}{a+b x} \, dx}{b c-a d}-\frac {d \int \frac {1}{c+d x} \, dx}{b c-a d}\\ &=\frac {\log (a+b x)}{b c-a d}-\frac {\log (c+d x)}{b c-a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 26, normalized size = 0.72 \[ \frac {\log (a+b x)-\log (c+d x)}{b c-a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.94, size = 26, normalized size = 0.72 \[ \frac {\log \left (b x + a\right ) - \log \left (d x + c\right )}{b c - a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.94, size = 46, normalized size = 1.28 \[ \frac {b \log \left ({\left | b x + a \right |}\right )}{b^{2} c - a b d} - \frac {d \log \left ({\left | d x + c \right |}\right )}{b c d - a d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 37, normalized size = 1.03 \[ -\frac {\ln \left (b x +a \right )}{a d -b c}+\frac {\ln \left (d x +c \right )}{a d -b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.13, size = 36, normalized size = 1.00 \[ \frac {\log \left (b x + a\right )}{b c - a d} - \frac {\log \left (d x + c\right )}{b c - a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.35, size = 25, normalized size = 0.69 \[ \frac {\ln \left (\frac {c+d\,x}{a+b\,x}\right )}{a\,d-b\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.44, size = 128, normalized size = 3.56 \[ \frac {\log {\left (x + \frac {- \frac {a^{2} d^{2}}{a d - b c} + \frac {2 a b c d}{a d - b c} + a d - \frac {b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{a d - b c} - \frac {\log {\left (x + \frac {\frac {a^{2} d^{2}}{a d - b c} - \frac {2 a b c d}{a d - b c} + a d + \frac {b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{a d - b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________