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 = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.81, size = 41, normalized size = 0.84 \[ \frac {{\left (b c - a d\right )} \log \left (b x + a\right ) - {\left (b c + a d\right )} \log \left (b x - a\right )}{2 \, a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 48, normalized size = 0.98 \[ \frac {{\left (b c - a d\right )} \log \left ({\left | b x + a \right |}\right )}{2 \, a b^{2}} - \frac {{\left (b c + a d\right )} \log \left ({\left | b x - a \right |}\right )}{2 \, a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 60, normalized size = 1.22 \[ -\frac {c \ln \left (b x -a \right )}{2 a b}+\frac {c \ln \left (b x +a \right )}{2 a b}-\frac {d \ln \left (b x -a \right )}{2 b^{2}}-\frac {d \ln \left (b x +a \right )}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 46, normalized size = 0.94 \[ \frac {{\left (b c - a d\right )} \log \left (b x + a\right )}{2 \, a b^{2}} - \frac {{\left (b c + a d\right )} \log \left (b x - a\right )}{2 \, a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 45, normalized size = 0.92 \[ -\frac {\ln \left (a+b\,x\right )\,\left (a\,d-b\,c\right )}{2\,a\,b^2}-\frac {\ln \left (a-b\,x\right )\,\left (a\,d+b\,c\right )}{2\,a\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 71, normalized size = 1.45 \[ - \frac {\left (a d - b c\right ) \log {\left (x + \frac {a^{2} d - a \left (a d - b c\right )}{b^{2} c} \right )}}{2 a b^{2}} - \frac {\left (a d + b c\right ) \log {\left (x + \frac {a^{2} d - a \left (a d + b c\right )}{b^{2} c} \right )}}{2 a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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