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 = 0.0076648, antiderivative size = 36, normalized size of antiderivative = 1., 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.
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Rule 36
Rule 31
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 = 0.0107835, size = 26, normalized size = 0.72 \[ \frac{\log (a+b x)-\log (c+d x)}{b c-a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 37, normalized size = 1. \begin{align*}{\frac{\ln \left ( dx+c \right ) }{ad-bc}}-{\frac{\ln \left ( bx+a \right ) }{ad-bc}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980413, size = 49, normalized size = 1.36 \begin{align*} \frac{\log \left (b x + a\right )}{b c - a d} - \frac{\log \left (d x + c\right )}{b c - a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71206, size = 58, normalized size = 1.61 \begin{align*} \frac{\log \left (b x + a\right ) - \log \left (d x + c\right )}{b c - a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.306326, size = 128, normalized size = 3.56 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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