Optimal. Leaf size=49 \[ \frac{(b c-a d) \log (a+b x)}{2 a b^2}-\frac{(a d+b c) \log (a-b x)}{2 a b^2} \]
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Rubi [A] time = 0.0367234, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {72} \[ \frac{(b c-a d) \log (a+b x)}{2 a b^2}-\frac{(a d+b c) \log (a-b x)}{2 a b^2} \]
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin{align*} \int \frac{c+d x}{(a-b x) (a+b x)} \, dx &=\int \left (\frac{-b c-a d}{2 a b (-a+b x)}+\frac{b c-a d}{2 a b (a+b x)}\right ) \, dx\\ &=-\frac{(b c+a d) \log (a-b x)}{2 a b^2}+\frac{(b c-a d) \log (a+b x)}{2 a b^2}\\ \end{align*}
Mathematica [A] time = 0.0079147, size = 37, normalized size = 0.76 \[ \frac{c \tanh ^{-1}\left (\frac{b x}{a}\right )}{a b}-\frac{d \log \left (a^2-b^2 x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 60, normalized size = 1.2 \begin{align*} -{\frac{\ln \left ( bx+a \right ) d}{2\,{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) c}{2\,ab}}-{\frac{\ln \left ( bx-a \right ) d}{2\,{b}^{2}}}-{\frac{\ln \left ( bx-a \right ) c}{2\,ab}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1113, size = 62, normalized size = 1.27 \begin{align*} \frac{{\left (b c - a d\right )} \log \left (b x + a\right )}{2 \, a b^{2}} - \frac{{\left (b c + a d\right )} \log \left (b x - a\right )}{2 \, a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29493, size = 90, normalized size = 1.84 \begin{align*} \frac{{\left (b c - a d\right )} \log \left (b x + a\right ) -{\left (b c + a d\right )} \log \left (b x - a\right )}{2 \, a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.29533, size = 71, normalized size = 1.45 \begin{align*} - \frac{\left (a d - b c\right ) \log{\left (x + \frac{a^{2} d - a \left (a d - b c\right )}{b^{2} c} \right )}}{2 a b^{2}} - \frac{\left (a d + b c\right ) \log{\left (x + \frac{a^{2} d - a \left (a d + b c\right )}{b^{2} c} \right )}}{2 a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.61764, size = 65, normalized size = 1.33 \begin{align*} \frac{{\left (b c - a d\right )} \log \left ({\left | b x + a \right |}\right )}{2 \, a b^{2}} - \frac{{\left (b c + a d\right )} \log \left ({\left | b x - a \right |}\right )}{2 \, a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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