Optimal. Leaf size=36 \[ \frac {\log (a+b x)}{b c-a d}-\frac {\log (c+d x)}{b c-a d} \]
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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}
\begin {gather*} \frac {\log (a+b x)}{b c-a d}-\frac {\log (c+d x)}{b c-a d} \end {gather*}
Antiderivative was successfully verified.
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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*}
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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.72 \begin {gather*} \frac {\log (a+b x)-\log (c+d x)}{b c-a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 37, normalized size = 1.03
method | result | size |
default | \(\frac {\ln \left (d x +c \right )}{a d -b c}-\frac {\ln \left (b x +a \right )}{a d -b c}\) | \(37\) |
norman | \(\frac {\ln \left (d x +c \right )}{a d -b c}-\frac {\ln \left (b x +a \right )}{a d -b c}\) | \(37\) |
risch | \(-\frac {\ln \left (b x +a \right )}{a d -b c}+\frac {\ln \left (-d x -c \right )}{a d -b c}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.38, size = 36, normalized size = 1.00 \begin {gather*} \frac {\log \left (b x + a\right )}{b c - a d} - \frac {\log \left (d x + c\right )}{b c - a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.50, size = 26, normalized size = 0.72 \begin {gather*} \frac {\log \left (b x + a\right ) - \log \left (d x + c\right )}{b c - a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 128 vs.
\(2 (26) = 52\).
time = 0.16, size = 128, normalized size = 3.56 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 46, normalized size = 1.28 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.26, size = 25, normalized size = 0.69 \begin {gather*} \frac {\ln \left (\frac {c+d\,x}{a+b\,x}\right )}{a\,d-b\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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